A318714 Denominator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3).

1, 5040, 50803200, 8449588224000, 442893616349184000, 55804595659997184000000, 315568291905804875857920000000, 211531737430299124385080934400000000, 6522145617145034649275530739712000000000, 254485460571619683408716971558739902464000000000

Offset

0,2

Comments

1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_3 is the third Bendersky constant.

a(n) is the denominator of b(n).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..134

Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Formula

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence

c_0 = 1, c_n = (-3/n) * Sum_{k=0..n-1} B_{2*n-2*k+4}*c_k/((2*n-2*k+1)*(2*n-2*k+2)*(2*n-2*k+3)*(2*n-2*k+4)) for n > 0.

a(n) is the denominator of c_n.

Example

1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(1 - 1/(5040*n^2) + 1513/(50803200*n^4) - 127057907/(8449588224000*n^6) + 7078687551763/(442893616349184000*n^8) - 1626209947417109183/(55804595659997184000000*n^10) + ... ).

Crossrefs

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).

Cf. A243263 (A_3).

Sequence in context: A008552 A221437 A221622 * A227669 A010800 A172544

Adjacent sequences: A318711 A318712 A318713 * A318715 A318716 A318717

Keyword

nonn,frac

Author

Seiichi Manyama, Sep 01 2018

Status

approved